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Bayesian Binary Search

Vikash Singh, Matthew Khanzadeh, Vincent Davis, Harrison Rush, Emanuele Rossi, Jesse Shrader, Pietro Lio

TL;DR

Bayesian Binary Search (BBS) reframes binary search as a probabilistic, density-guided process that uses a learned search-space PDF to pick split points. By estimating $p(x)$ with supervised or unsupervised methods and updating the PDF after each probe, BBS achieves faster convergence on nonuniform distributions and remains robust to uniform cases. The authors validate BBS on simulated distributions (normal, bimodal, exponential) and a real-world Lightning Network channel probing task, showing substantial reductions in required probes versus classical binary search, with performance contingent on PDF accuracy. They discuss limitations (PDF-estimation overhead, drift) and outline future directions, including adaptive PDF methods and broader domain applications, underscoring the practical potential of integrating probabilistic learning into fundamental search algorithms.

Abstract

We present Bayesian Binary Search (BBS), a novel probabilistic variant of the classical binary search/bisection algorithm. BBS leverages machine learning/statistical techniques to estimate the probability density of the search space and modifies the bisection step to split based on probability density rather than the traditional midpoint, allowing for the learned distribution of the search space to guide the search algorithm. Search space density estimation can flexibly be performed using supervised probabilistic machine learning techniques (e.g., Gaussian process regression, Bayesian neural networks, quantile regression) or unsupervised learning algorithms (e.g., Gaussian mixture models, kernel density estimation (KDE), maximum likelihood estimation (MLE)). We demonstrate significant efficiency gains of using BBS on both simulated data across a variety of distributions and in a real-world binary search use case of probing channel balances in the Bitcoin Lightning Network, for which we have deployed the BBS algorithm in a production setting.

Bayesian Binary Search

TL;DR

Bayesian Binary Search (BBS) reframes binary search as a probabilistic, density-guided process that uses a learned search-space PDF to pick split points. By estimating with supervised or unsupervised methods and updating the PDF after each probe, BBS achieves faster convergence on nonuniform distributions and remains robust to uniform cases. The authors validate BBS on simulated distributions (normal, bimodal, exponential) and a real-world Lightning Network channel probing task, showing substantial reductions in required probes versus classical binary search, with performance contingent on PDF accuracy. They discuss limitations (PDF-estimation overhead, drift) and outline future directions, including adaptive PDF methods and broader domain applications, underscoring the practical potential of integrating probabilistic learning into fundamental search algorithms.

Abstract

We present Bayesian Binary Search (BBS), a novel probabilistic variant of the classical binary search/bisection algorithm. BBS leverages machine learning/statistical techniques to estimate the probability density of the search space and modifies the bisection step to split based on probability density rather than the traditional midpoint, allowing for the learned distribution of the search space to guide the search algorithm. Search space density estimation can flexibly be performed using supervised probabilistic machine learning techniques (e.g., Gaussian process regression, Bayesian neural networks, quantile regression) or unsupervised learning algorithms (e.g., Gaussian mixture models, kernel density estimation (KDE), maximum likelihood estimation (MLE)). We demonstrate significant efficiency gains of using BBS on both simulated data across a variety of distributions and in a real-world binary search use case of probing channel balances in the Bitcoin Lightning Network, for which we have deployed the BBS algorithm in a production setting.
Paper Structure (34 sections, 11 equations, 7 figures, 5 tables, 5 algorithms)

This paper contains 34 sections, 11 equations, 7 figures, 5 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. Classical Binary Search/Bisection
  4. Bayesian Binary Search (BBS)
  5. Problem Formulation
  6. Search Space Probability Density Function Estimation
  7. Supervised Learning Approaches
  8. Unsupervised Learning Approaches
  9. Binary Search with Probabilistic Bisection
  10. Experiments on Simulated Data
  11. Experimental Setup
  12. Distribution Parameters
  13. Target Values Generation
  14. Search Procedure
  15. Metrics Collected
  16. ...and 19 more sections

Figures (7)

  • Figure 1: Normal Distribution: Basic vs. BBS Convergence Comparison
  • Figure 2: Lightning Probing Experiment: Basic vs. BBS Convergence Comparison
  • Figure 3: Basic vs BBS Convergence with KLD Separation of Estimated Search Space PDF and Actual PDF
  • Figure 4: Bimodal Distribution: Basic vs. BBS Convergence Comparison
  • Figure 5: Exponential Distribution: Basic vs. BBS Convergence Comparison
  • ...and 2 more figures